3.11 \(\int \frac{\sqrt{c+d x} (e+f x)}{x} \, dx\)

Optimal. Leaf size=54 \[ 2 e \sqrt{c+d x}-2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\frac{2 f (c+d x)^{3/2}}{3 d} \]

[Out]

2*e*Sqrt[c + d*x] + (2*f*(c + d*x)^(3/2))/(3*d) - 2*Sqrt[c]*e*ArcTanh[Sqrt[c + d
*x]/Sqrt[c]]

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Rubi [A]  time = 0.0727782, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ 2 e \sqrt{c+d x}-2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\frac{2 f (c+d x)^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[c + d*x]*(e + f*x))/x,x]

[Out]

2*e*Sqrt[c + d*x] + (2*f*(c + d*x)^(3/2))/(3*d) - 2*Sqrt[c]*e*ArcTanh[Sqrt[c + d
*x]/Sqrt[c]]

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Rubi in Sympy [A]  time = 7.95638, size = 49, normalized size = 0.91 \[ - 2 \sqrt{c} e \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )} + 2 e \sqrt{c + d x} + \frac{2 f \left (c + d x\right )^{\frac{3}{2}}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x+e)*(d*x+c)**(1/2)/x,x)

[Out]

-2*sqrt(c)*e*atanh(sqrt(c + d*x)/sqrt(c)) + 2*e*sqrt(c + d*x) + 2*f*(c + d*x)**(
3/2)/(3*d)

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Mathematica [A]  time = 0.0763735, size = 53, normalized size = 0.98 \[ \frac{2 \sqrt{c+d x} (c f+3 d e+d f x)}{3 d}-2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[c + d*x]*(e + f*x))/x,x]

[Out]

(2*Sqrt[c + d*x]*(3*d*e + c*f + d*f*x))/(3*d) - 2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x
]/Sqrt[c]]

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Maple [A]  time = 0.011, size = 46, normalized size = 0.9 \[ 2\,{\frac{1}{d} \left ( 1/3\,f \left ( dx+c \right ) ^{3/2}+de\sqrt{dx+c}-\sqrt{c}de{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)*(d*x+c)^(1/2)/x,x)

[Out]

2/d*(1/3*f*(d*x+c)^(3/2)+d*e*(d*x+c)^(1/2)-c^(1/2)*d*e*arctanh((d*x+c)^(1/2)/c^(
1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(f*x + e)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224652, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{c} d e \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (d f x + 3 \, d e + c f\right )} \sqrt{d x + c}}{3 \, d}, -\frac{2 \,{\left (3 \, \sqrt{-c} d e \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) -{\left (d f x + 3 \, d e + c f\right )} \sqrt{d x + c}\right )}}{3 \, d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(f*x + e)/x,x, algorithm="fricas")

[Out]

[1/3*(3*sqrt(c)*d*e*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(d*f*x + 3*
d*e + c*f)*sqrt(d*x + c))/d, -2/3*(3*sqrt(-c)*d*e*arctan(sqrt(d*x + c)/sqrt(-c))
- (d*f*x + 3*d*e + c*f)*sqrt(d*x + c))/d]

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Sympy [A]  time = 6.31306, size = 110, normalized size = 2.04 \[ - 2 c e \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x \wedge - c < 0 \end{cases}\right ) + 2 e \sqrt{c + d x} + \frac{2 f \left (c + d x\right )^{\frac{3}{2}}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x+e)*(d*x+c)**(1/2)/x,x)

[Out]

-2*c*e*Piecewise((-atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sqrt(c
+ d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c < c + d*x)), (atanh(sqrt(c + d*x)/sqrt(c
))/sqrt(c), (-c < 0) & (c > c + d*x))) + 2*e*sqrt(c + d*x) + 2*f*(c + d*x)**(3/2
)/(3*d)

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GIAC/XCAS [A]  time = 0.215867, size = 77, normalized size = 1.43 \[ \frac{2 \, c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) e}{\sqrt{-c}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} d^{2} f + 3 \, \sqrt{d x + c} d^{3} e\right )}}{3 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)*(f*x + e)/x,x, algorithm="giac")

[Out]

2*c*arctan(sqrt(d*x + c)/sqrt(-c))*e/sqrt(-c) + 2/3*((d*x + c)^(3/2)*d^2*f + 3*s
qrt(d*x + c)*d^3*e)/d^3